Problem: Let $f(x)=e^{x^2+2x}$ Where does $f$ have critical points? Choose all answers that apply: Choose all answers that apply: (Choice A) A $x=-2$ (Choice B) B $x=-1$ (Choice C) C $x=0$ (Choice D) D $f$ has no critical points.
A critical point of $f$ is a point in the domain of $f$ where the derivative is either equal to zero or undefined. So in order to find the critical points of $f$, let's find its derivative. $\begin{aligned} f'(x)&=\dfrac{d}{dx}\left[ e^{x^2+2x} \right] \\\\ &=e^{x^2+2x} \cdot \dfrac{d}{dx}[x^2+2x] \\\\ &=(2x+2)e^{x^2+2x} \\\\ &=2(x+1)e^{x^2+2x} \end{aligned}$ Now let's look for $x$ -values where $f'$ is zero or undefined. $2(x+1)e^{x^2+2x}=0$ at $x=-1$. $2(x+1)e^{x^2+2x}$ is never undefined, so $f'$ is never undefined. In conclusion, this is the only $x$ -value where $f$ has a critical point: $x=-1$